Chapter 4.3, Problem 33ES

Use the properties of even and odd integers that are listed in example 4.3.3 to do exercise 33 and 34.
33. When expressions of the form $\left(x-r\right)\left(x-s\right)$ are multiplied out, a quadratic polynomial is obtained. For instance, $\left(x-2\right)\left(x-\left(-7\right)\right)=\left(x-2\right)\left(x+7\right)=x^2+5x-14$ .
a. What can be said about the coefficients of the polynomial obtained by Multiplying out $\left(x-r\right)\left(x-s\right)$ when both r and s are odd integers? When both r and s are even integers? When one of r and s is even and the other is odd?
b. It follows from part (a) that $x^2-1253x+255$ cannot be written as a product of two polynomials with integer coefficients. Explain why this is so.